Modeling Diffusive Mixing in Antisolvent Crystallization

Diffusion controls local concentration profiles at interfaces between segregated fluid elements during mixing processes. This is important for antisolvent crystallization, where it is intuitively argued that local concentration profiles at interfaces between solution and antisolvent fluid elements can result in significant supersaturation overshoots over and above that at the final mixture composition, leading to poorly controlled nucleation. Previous work on modeling diffusive mixing in antisolvent crystallization has relied on Fickian diffusion, where concentration gradients are the driving force for diffusion. This predicts large overshoots in the supersaturation at interfaces between solution and antisolvent, as is often intuitively expected. However, chemical potential gradients provide a more physically realistic driving force for diffusion, and in highly nonideal solutions, such as those in antisolvent crystallization, this leads to nonintuitive behavior. In particular, as solute diffusion toward antisolvent is severely hindered, it can diffuse against its concentration gradient away from antisolvent. We apply thermodynamically consistent diffusion model based on the multicomponent Maxwell–Stefan formulation to examine diffusive mixing in a nonideal antisolvent crystallization system. Large supersaturation overshoots above that at the final mixture composition are not found when a thermodynamically consistent approach is used, demonstrating that these overshoots are modeling artifacts and are not expected to be present in physical systems. In addition, for certain conditions, localized liquid–liquid spinodal demixing is predicted to occur during the diffusive mixing process, even when the final mixture composition is outside the liquid–liquid phase separation region. Intermittent spinodal demixing driven by diffusive mixing may provide a novel explanation for differences of nucleation behaviors among various antisolvents.


Methodology Model system
To model the diusive mixing in anti-solvent crystallisation, a ternary component system consisting of water, ethanol and glycine is considered. In this system, glycine is the solute, water is the solvent and a mixture of ethanol/water was used as the anti-solvent. The system was modelled as a static channel, with one part lled with aqueous glycine solution, and the other part the anti-solvent/solvent mixture. Figure 1 describes an example of the initial volume fraction prole of the channel. Note the volume fraction refers to φ i = V i ·x i  This was for a 50:50 initial ratio of solution to antisolvent within the channel. Antisolvent composition was 83.5% ethanol and 16.5% water.

Diusion Coecients
The Fickian mutual diusion coecients for Glycine-water 16 and ethanol-water 711 mixtures, are shown in gure2.

Time-Step checks
The time-steps used in the simulations were 0.001s and 0.1s. To check this did not eect the accuracy of the results, simulations were performed for time-steps of 0.001s and compared to 0.1s. Figure 3 shows the volume fraction proles at short times. Figure 3: Comparison of volume fraction (φ) proles for time steps 0.1 and 0.001s. Pure ethanol was used as the antisolvent, the solution to antisolvent ratio was 1:1 within the channel and the initial supersaturation was 0.9. Dotted lines indicate 0.1 and solid lines represent 0.001s. The non-ideal solution model was used.   To capture earlier behaviour accurately, the time step used was 0.001s. At 20s this was reduced to 0.1s as no dierences in mass fraction proles were present.

Ideal parametric study
To further develop understanding of diusive mixing in antisolvent crystallisation, a parametric study was performed over three key process parameter. These were antisolvent compo- sition, ratio of solution to antisolvent and the initial supersaturation. Figure 5 summarises the results of this study. To quantify the comparisons across the parameter values, the maximum supersaturation attained during mixing is plotted along with the fully mixed supersaturation. This gives insight into how nucleation conditions are impacted by the relative parameter, along with the relative size of the 'overshoot' in supersaturation. It also highlights that mixing should be carefully considered in the design of crystallisation processes, as this inuences crystallisation outcomes. The movement of the peak supersaturation into the antisolvent is present across all parameters, hence this was not considered in this analysis.
As one parameter is varied, the others are held constant at 'centre point' values. Table 1 summarises this. Figure 5a shows how the peak supersaturation is eected by initial antisolvent compo- greater supersaturations. The trend in the ratio of peak to nal supersaturation is more apparent in gure 5b. Larger overshoots are present in the pure antisolvent, which becomes increasingly lower as the water content increases. From intuition this is expected. In the ideal model, supersaturation reduces to the ratio of mol fraction of glycine and the saturated value, i.e. solubility. Glycine diuses down its composition gradient and the overshoot in supersaturation will be dictated by the local solubility of the antisolvent. Solubility is extremely low in the pure antisolvent and thus large overshoots are predicted. In many cases, these overshoots are undesirable, for example eective secondary nucleation processes require control over supersaturation. High levels of supersaturation can lead to primary nucleation, resulting in unwanted crystal properties. For antisolvents that have higher water content, the peak supersaturation decreases with respect to the nal value. This may help in controlling nucleation. The driving force for crystallisation is supersaturation, so an antisolvent composition should be selected to balance control and to achieve a suitable nucleation rate. Figure 5c is the eect of varying the solution:antisolvent ratio. The peak supersaturation at early times was found to be independent of this ratio. This agrees with the study by Thorson 12 as discussed in main paper.. In terms of diusion, this can be explained by considering short time behaviour at the interface. Immediately after the onset of mixing glycine diuses into the antisolvent and an overshoot in supersaturation is observed. At distances away from the interface, the glycine does not start to diuse until there is a driving force present. i.e. the compositional gradient will be at within regions away from the interface at short times. As diusion proceeds at the interface a compositional gradient is generated, as glycine moves into the antisolvent. The same reasoning can be applied to the diusion of water and ethanol. At long times, the ratio of solution:antisolvent impacts the supersaturation prole, which can be seen in the nal supersaturation. A straight forward relationship is seen, with more antisolvent leading to higher nal supersaturations. This is simply due to the lower solubility of the solvent mixture. One limitation of this study is the exclusion of nucleation. Nucleation would act to lower the supersaturation, and if it is suciently high at the interface, then long term predicted proles could fail to represent the physical system. Quantitative behaviour however, would still be expected to remain reasonably accurate.
The last parameter investigated was the initial supersaturation, which is essentially equivalent to the initial mass fraction of glycine in the solution. The trends in both peak and nal supersaturation are once more what would be expected from intuition. That is, increased glycine in the initial solution results in an increased supersaturation in the fully mixed system. The peak supersaturation follows the same trend. This is caused by the increased driving force for the diusion of glycine. This means more glycine would be present in the antisolvent thus, greater degree of supersaturation.

Non-ideal parametric study
The eects of key processes parameters were studied for mixing in ideal solutions, however the previous sections have shown that the ideal model does not represent the physical process.
To correct this, the activity gradient was considered as the driving force. Figure 6 details the eect of these parameters.
Antisolvent composition (6b) indicates that the peak supersaturation experienced is only slightly greater than the nal value. Additionally the 'overshoot' is no longer present initially after mixing as is with the ideal case, but occurs at the antisolvent channel wall towards the end of mixing. This suggests that large overshoots in supersaturation as expected through intuition do not exist. However, gure 6a highlights that the initial peak supersaturation increases with increasing ethanol wt % in the antisolvent, similarly to the ideal case, as does the value of the maximum supersaturation attained during mixing.
The ratio of solution:antisolvent again produced results similar to the ideal model. For ratios of 1:1 or greater the peak supersaturation is approximately constant. These ratios the initial peak is located at the interface between the solution and antisolvent, and was found to be independent of solution:antisolvent ratio. At small time scales, mixing is only experienced at the interface and this is expected. Low amounts of relative antisolvent lead to the nal supersaturation being relatively low, and the initial peak is found to be the highest in magnitude. For ratios less than one, the peak value is realised due to high antisolvent composition in the nal mixture. For the ratio of 1:3 (solution:antisolvent) the largest overshoot occurs as the antisolvent mixes into the solution. Glycine moves relatively slow in comparison to the water/ethanol intermixing and hence supersaturation is generated as ethanol lowers local solubility at the solution wall.
Quantitative trends were unchanged from the ideal and non-ideal models with higher initial supersaturation (initial glycine mass fraction) producing higher supersaturations. The key dierence between models is once more the magnitude of the overshoots. In the non-ideal model these are small, and increase with higher supersaturations as more glycine is present.
Spatiotemporal proles are shown in the When comparing trends of key process parameters for ideal and non-ideal models we nd they are similar. However a key dierence is observed when considering the magnitudes of peak supersaturations. The non-ideal model predicts more modest peaks. The composition of the peak varies between models and therefore signicantly dierent nucleation conditions are modelled. The ideal model therefore fails to predict real-life physical process, even if the eects of key process parameters make sense intuitively.